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DESCRIPTION OF THE GIANT ANGLE DIPOLE
R. Hilton
To cite this version:
R. Hilton. DESCRIPTION OF THE GIANT ANGLE DIPOLE. Journal de Physique Colloques, 1984, 45 (C6), pp.C6-255-C6-264. �10.1051/jphyscol:1984630�. �jpa-00224232�
JOURNAL DE PHYSIQUE
Colloque C6, supplément au n06, Tome 45, juin 1984 page C6-255
DESCRIPTION OF THE G I A N T ANGLE D I P O L E
R.R. Hilton
Physies Department, Technieu?, University Munich, 8046 Garching, F.R.G.
Résumé - Un analogue angulaire de la résonance géante dipolaire ("Giant Angle ~i~ole")' est décrit et étudié pour le cas d'un grand nombre de particules aux spins saturés dans un oscillateur déformé.
Le caractère du mode collectif de basse énergie qui correspond à cette excitation est déterminé comme rotation des protons contre les neutrons plus une petite oscillation de forme au volume conservé.
Selon le modèle, le rapport des énergies d'excitation du mode dipolaire angulaire et de la résonance géante dipolaire est égal à la déformation.
La valuer BM(1) est calculée et les effets des corrélations sont estimés. Un court exposé critique d'autres études théoriques est présenté.
Abstract - An angular analogue to the Giant Dipole (the Giant Angle Dipole)+ is described and investigated fot the case of a spin saturated
large particle number system contained in a deformed oscillator well.
The character of the low energy collective mode corresponding to this excitation in determined as a sum of contra proton-neutron rotation plus small volume conserving shape oscillations. The mode1 predicts the Angle Dipole to Giant Dipole excitation energy ratio as equal to the deformation. The BM(I) rate is calculated and the effects of correla- tions are assesed. A brie£ critical review of other theoretical work is given.
INTRODUCTION
By its very nature nuclear structure i s concerned with t h e unenviable task of studying a system for which no fully microscopic description can be implemented. On the other hand nuclei manifest facets of t h e nuclear many body problem, t h e study of which does not require t h e detailed knowledge that a complete microscopic treatment would supply. Such an area of interest i s represented by those dynamical States of the nucleus which may be described in terms O+
simple collective vibrational modes of excitation. Perhaps t h e simpleat and most prominent example i s that of t h e Giant Dipole resonance which may be visulized a s vibrations of t h e relative distance between the centres of mass of t h e neutrons and protons. Here we shall consider its angular analogue, t h e Giant Angle Dipole /1/. Such motion may be
roughly described a s excitations of a deformed nucleus in which t h e
+ ~ h e name "Angle Dipole" was coined by A.M.Lane, whose spontaneous christening, during the course of a discusion held in summer 1971, has been kept. In t h e author's opinion its obvious pictorial quality more than compensates for any momentary doubt about the quantum numbers which should be assigned t o t h i s state.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984630
proton and neutron densities vibrate against one another in angular fashion about an axis perpendicular t o t h e symmetry axis of the
system. In section 1 we inéroduce the collective description of t h i s mode generated from the single particle description that one obtains +rom t h e shell model. In section 2 we give t h e neccessary and
sufficient conditions of t h e model and establish the inertial
parameter and restoring force constant required in this description.
In t h e following section w e apply these criteria t o the case of independent fermions in a harmonic well and find explicit expressions for the appropriate collective coordinate and momentum, together with the energy and transition strength of this mode. Having establish the character of the motion in section S we next assess the eff ects of correlations on t h e energy and BM(1) value. In section 7 t h e present experimental situation i s considerrd and in t h e last section w e take a quitk and jaundiced look at other treatments of this state which have a s yet emerged in t h e literature.
1-DESCRIPTION OF THE MODE
The present investigation i s directed towards heavy deformed even- even nuclei. Our aim has been t o generate +rom the highly successful single particle picture that one obtains from t h e shell model, a collective description in which we may display states of t h e system in which t h e nucleus participates coherently.
10 t h i s end we introduce collective coordinates 3 and A and a remaining set of variables Cq, ,p, > commuting with 5 and A which describe the remaining degrees of freedom in t h e system. T h e shell model Hamiltonian written in terms of the new variables in general takes t h e iorm
Here t h e first term describes collective vibrations, t h e second depicts the motion of al1 remaining degrees of freedom and H b u p represents possible coupling between such motions. Since w e want t o identify the first term with oscillations about some equilibrium shape it would be most reasonable if J and A could be chosen !BO that t h e coupling term be very small and collective motion described by
harmonic motion, in the same spirit a s that for t h e Giant Dipole /2/
i.r.
Here B and C represents t h e effsctive inertial paramrter and stiifness constant appropriate t o this motion. Such a Hamiltonian would have a ground state of t h e product form @,iq)X,(S) and it would then be natural t o interpret wave functions of t h e form @,<q)X,(J) as representing collective excitations of th= system.
The inertial parameter B and stiffness constant C appropriate t o the motion may then br established t o take t h e form
and
C = ~(<MB>-<N'><H>) respect i vel y.
2-CRITERIA FOR THE MODEL
Unfortunately such a description is in general greatly complicatad by the problem of def ining explicit expressions for t h e collective variables. As a rule they leave large residual coupling terms after the transformation t o the new variables and are therefore inappropriate for Our purpose. Thus t h e first question we are forced t o ask is, what are the conditions under which a Hamiltonian will exhibit such a simple form Ho? More precisely stated, what are t h e neccessary and sufficient conditions for the shell model Hamiltonian H, t o exhibit t h e decoupled harmonic structure ( 2 ) ?
The weakest form that has been found t o date i s a s followsr If Y, represents the normalized ground state of t h e shell model Hamiltonian H, and a real c-number  and hermitian collective operators
ïï and 5 can be found such that the following relations are satisfied
and the commutator
where f i s in general a q-number (operator) satisfying t h e conditions
+or some sequence of eigenstates of H,CYn> n=011,2 etc. sa that
We then gain a sequence of related theorems and corrollaries which prove that if these conditions are met t h e Hamiltonian H takes t h e simpler decoupled (harmonic) form Ho, with corresponding excited states of t h e product form Bo(q)X,(3). In view of Our space limitations we shall not have the opportunity in this account t o present these
existence theorems in their entirety but will have t o content ourselvea with their statement. Full details will be included in a forthcoming paper .
Theorem 1 - A Y, satisfying ( 5 ) - ( 7 ) is an eigenfunction of t h e operator Ji Z + h2 with eigenvalue X.
Corrollary 1 - The eigenvalue  i s unique.
Corrollary II - The ground state eigenfunction of H o , i s
proportional t o Te.
Corrollary III - The ground state kinetic energy of collective motion equals the potential energy.
Theorem II - A shell model Hamiltonian H, whose ground state wave function satisfies ( 5 ) - ( 7 ) has t h e decoupled structure Ho,
where Yotakes the form B,(qiXo(3> in which B,(q) and X e (3) represent t h e lowest eigenstates of Hc,, , and H n respective1 y.
Theorem III - If (5)-(7) are fulfilled together with t h e condition that f, (n=1,2 etc.) are equal t o unity, eigenstates of H exist of t h e form $,(q)X,(S) representing excited states of collective motion.
3-CHOICE OF COLLECTIVE VARIABLES
The question which naturally arises is, how does one generate proper collective variables for the system? In order t o gain from t h e single particle picture a collective description we search for a set of operators CA: > , which satisf y,
from which one gains t h e matrix M. We proceed by dinding the eigenvectors of MT i.e.
and construct t h e operators G!* t
The appropriate coordinates and momenta are then given a s
J = :+lC( Q,)/2c, and Ji = (Q,- t Q,)/2ic, 112)
in which c, and c a are normalization constants fixed by t h e commutation relation of J and Ji and t h e condition
It should perhaps be emphasised at this point that the commutator of 3 and JI will in general turn out t o be an operator and i s t h e reason we have had t o be s o careful t o establish t h e properties of the collective description through the medium of existence theorems.
4-DEFORMED OSCILLATOR
For t h e case of fermions moving in a prolate deformed iwz<wl) oscillator well of t h e +orm
collective variable ratisf ying the neccessary and suf f icient conditions of t h e mode1 may be constructed for a system corresponding t o a heavy even-even dei ormed nucleus. The expl ici t f orms of the Giant fingle Dipole collective coordinate and momentum for a system comprising Z protons and N neutrons are:
in which represents t h e total energy 2-component of t h e (protons o r neutrons) and
in which B = <H,>N/ (<H,>z + < H ~ > N ) (17) It i s perhaps worth noting here that the appearance of explicit momentum dependence in t h e coordinate i s really a reflection of its non-locality, a property which i s t o be expected from such a RPA based description.
T h e nature of t h e mode i s established directly from (13) and for a non self-conjugate niicleus (N#Z) can be seen t o be a positive parity
quadrupole e x c i t a t i o n c a r r y i n g i s o s p i n T = l ( w i t h small T=O admixture), and comprises a 1 i n e a r combination of angular momentum z - p r o j e c t i o n M i e l . For a self-conjugate nucleus t h e mode r e v e r t s t o a pure T=1 s t a t e .
The e x c i t a t i o n energy of t h e Giant Angle Dipole (Ea.d.) i s given from the tzommutator
Expressed i n a l e s s mode1 dependent fashion as a r a t i o t o t h a t f o r t h e Giant D i p o l e (Eg.d. ) i t takes t h e form
whereby 6 4 s t h e Nilsson deformation parameter. The r e s u l t immediately suggests t h a t we may expect t h i s mode t o have a low e x c i t a t i o n energy, t y p i c a l l y 3-4 Mev f o r deformed n u c l e i i n t h e r a r e e a r t h region.
O f i n t e r e s t a l s o are t h e electromagnetic t r a n s i t i o n p r o b a b i l i t i e s as they g i v e i n s i g h t i n t o t h e e x c i t a t i o n mechanism t h a t can be used t o e s t a b l i s h t h e mode. Although t h e Giant Angle D i p o l e may be p a r t i a l l y e x c i t e d by quadrupole r a d i a t i o n , from (16) i t i s c l e a r l y most s t r o n g l y e x c i t e d by magnetic d i p o l e components. The t r a n s i t i o n s t r e n g t h may be estimated from an e v a l u a t i o n o f t h e reduced M l t r a n s i t i o n p r o b a b i l i t y BtMl), given by,
Here gp and g, represent t h e o r b i t a l proton and neutron gyromagnetic r a t i o s r e s p e c t i v e l y . No s p i n c o n t r i b u t i o n occurs as we are d e a l i n g w i t h a s p i n saturated system.
For a t y p i c a l w e l l deformed r a r e e a r t h nucleus e.g15b~d f o r which 6=0.251 N=92, 2164 we gain a B(M1) value
From (20) i t seems p o s s i b l e t h a t higher t r a n s i t i o n s t r e n g t h s may be found i n heavier deformed n u c l e i i n the a c t i n i d e region.
5-CHARACTER OF THE MODE
The character o f t h e mode may be ascertained by l o o k i n g more c l o s e l y a t t h e f orm of t h e generator ILD.. We may decompose t h e e s s e n t i a l operator appearing i n &.D. as
The f i r s t term produces a r o t a t i o n w h i l s t t h e second generates two orthogonal shears. For small displacements about t h e e q u i l i b r i u m shape t h e Giant Angle Dipole i s thus seen t o be made up of a sum o f two motions: a c o n t r a angular r o t a t i o n a l o s c i l l a t i o n of t h e proton and neutron d e n s i t i e s together w i t h a small admixture of two orthogonal shears, producing volume conserving deformation v i b r a t i o n s . The form of
the motion i s illustrated in Fig.1 below.
Fig. 1
It must be emphasised that t h i s contrasts sharply with t h e picture of pure contra rotation which classical intuition might suggest, a s t h e generators of such motion lead t o an excited collective state made up of linear combinations of single particle states from different osci 11 ator shel 1 s i . e.
where Lz,Ln represent t h e total proton and neutron angular momentum component perpendicular t o t h e symmetry axis respective1 y. This i s, from the point of view of t h e coherence of such a 1ow energy state, a physically unacceptable description. The Giant Angle Dipole generator ensures such incoherent mixtures d o not occur.
6-EFFECTS OF CORRELATIONS
In t h i s section we shall make a first attempt t o account for the effects of the short and long range correlations on bath t h e energy and transition rates. Strictly speaking t h i s i s outside the domain of our rigorous mode1 and therefore some of the statements will neccessarily be rather rough. However t h e arguments d o give t h e general trends of these effects and allow u s t o establish absolute rather than relative excitation energies.
In terms of t h e inertial parameter B and stiffness constant C of t h e motion t h e excitation energy and transition rate take the form
We may simulate t h e ef fects of long range isovector forces by postulating an additional restoring force not contained in t h e shell mode1 container well proportional t o the difference between t h e
displaced and undisplaced wavefunction overlap. Thus for a displacement 3 the restoring force is
In which JI i s the generator appropriate t o t h e motion. T o extract N, we look t o t h e Giant Dipole t o establish t h e extra restoring f o r c e
required t o gain t h e experimentally found energy, which has t h e form
with Ne= 5.6~'. Here p L i s t h e relative momentum generating
displacements between t h e proton and neutron centres of mass. The resulting energy then takes t h e form
JC (C+C ' /C 3: Eosc gi ving
JC~+Z.~W~/(W,.G) I.EOSC 1 - 3.4.1~0s~
Although attractive because of its self-containment this idea i s conceptually faulty since it has tacitly assumed that universal forces are at play. As we have already been made aware t h e Giant Angle Dipole excitation lies within a P N = 0 subspace, whilst the Giant Dipole
excitation exists in t h e A N = l subspace. It i s therefore t o be expected from quite general considerations /3/ that t h e effective forces acting in the t w o subspaces will be different. We t h u s have t o appeal t o studies of t h e interactions in rertricted subspaces t o ascertain the effective coupling constants. Such studies /4/ suggest that t h e L N = O isovector t o isoscaler coupling constant ratio magnitude is of order 0.6. This would imply that such effects raise the energy and transition rate by with respect t o t h e oscillator value.
In order t o asses the effects of short range correlations and a more realistic container well we evaluated B and C for the case of nucleons in a Nilsson potential subject t o pairing interactions. Al1 single particle levels of t h e N=4,5 proton and N=5,6 neutran shells were included in t h e calculation. For t h e case of '5bGd t h e following results emerged. Due t o pairing the resul ting inertial parameter (3) evaluated using a BCS wave function of the f orm,
showed a reduction with respect t o the oscillator value of 2.3 i .e.
However the resulting excitation energy was increased by only a factor 1.2 with respect t o the oscillator value i.e. E = 1.2.Eosc.
On collecting these results together we f ind for t h e Giant Angle Dipole excitation energy i n t S b G d
Ea.d. * 2.4- " 3 Nev
For the B(M1) transition strength w e gain t h e value
Taking account of deviations of t h e proton and neutron gyromagnetic ratios from their non interacting values of 1 and O respectively can produce a futher 10% reduction in this value.
7-THE EXPERIMENTAL SITUATION
Recently a high resolution search O+ the <e,e') scattering spectrum of 'Sb~d, perf ormed at t h e Darmstadt electron 1 inear accelerator ,
revealed an interesting resonance at 3.075 Mev.
Shown in Fig. 2 i s a background subtracted (e,e') low electron incident energy backward angle spectrum.
' 5 6 ~ d (esen) E0.30MeV
1.5 2.0 2.5 3.0 3.5 I
E x c i t a t i o n Energy ( M e V )
Fig.2 Ce,&') excitation spectrum of I b b ~ d taken from Bohle et al / 5 /
The figure shows a unif orm excitation of many low spin ipresumably 2 quasi-particle) levels apart from one strongly excited 1* state. The only other state having any strength i s a well knowc 3' . The observed energy of this l+ agrees well with that expected from a Giant Ongle Dipole excitation. Form factor behaviour al s o f avours this assignment.
A search of the ie,e') spectrum i n I C b ~ d , which is not a good rotor, failed t o reveal such a state in the excitation spectrum, indicating again that we a r e dealing with a state of an essentially collective character. However the BiM1) value extracted ( 1.3 kt) shows a marked discrepancy with that gained from Our collective description (4.6 k.:) At first sight the experimental value appears little different t o that for the single particle estimate and were it not for the other
corroborating evidence one could well believe this state not t o have a collective origin. It must be remembered that we are not dealing with spin-flip processes here which are subject t o quenching. This puzzling feature has very recently begun t o resolve itself. Measurements on seven further rare rarth nucleir have shown the systematic presence of such collective 1+ States. In some cases they reveal level splittings which we are not able t o describe with t h e present model. However preliminary values for t h e total summed BiM1) strength f ound in one of the nuclei e ~ a m i n e d ' ~ ~ ~ ~ , with which we should compare Our total transition strength, amount t o almost 2.6 kt, and i s much closer t o our e x p e c t e d l B b ~ d value of 4.6 v:. This suggests that i n l b b ~ d a substantial fraction of the strength has been lost in the background.
This i s an exciting development since a value above the single
particle estimate bath supports the collective nature of the state and points t o t h e fact that t h e puzzlingly small measured strength inlbbGd has really given a false first impression. The experimentally found excitation levels show a mean energy behaviour following an 6''~. 8 variation, in conformation with that expected from such a collective state. Certainly the recent findings give hope that this collective description i s on the right track.
*D. 80hle private communication
8-CRITICAL REVIEW
In t h i s final section we give a brief and mostly jaundiced virw of other studies of this mode which have or are about t o appear in t h e literature. We shall therefore concentrate our attention on t h e weaknesses or inconsistencies that in Our view will require attention in any future work. T o begin w e make t w o overall criticisms which affect t h e majority of these investigations. First t h e a priori use of pure proton-neutron contra rotation generators t o describe t h i s motion /6,7,9,10,11,14,15/ which, a s we have already indicated, quantum mechanically does not describe a coherent low energy state. Secondly t h e restriction of the treatments t o N=Z nuclei /6,8,9,10,11,12/.
In /6/ a study built on an angular analogue t o t h e Giant Dipole analysis of Goldhaber-Teller was presented. Apart from the coherrnce difficulties associated with using the isovector angular momentum generator t h e analysis tacitly assumes the effective forces acting in t h e case of relative centres of mass displacement are t h e s a m e a s those for rotational motion. This i s conceptually at fault and inevitably leads t o t o o high an energy. However t h e unrealistic high inertial parameter value chosen leads t o a suppression of t h i s effect. The original restriction t o self-conjugate nuclei was ostensibly lifted in /7/, however t h e collective variables introduced in t h e description are inappropriate for t h e N # Z case and d o not decouple t h e mode from the rotational motion.
In /8/ a sum rule approach together with a scaling model argument was used t o relate t h e excitation energy of t h e mode t o that of t h e i soscal er quadrupol e vibrations. However t h e fi na1 generator f ound produces excited collective States comprising linear combinations of
&=O and &=2, which i s therefore inconsistent with t h e description of
a coherent low energy state. Also t h e rigid body inertial parameter of the total system was used instead of t h e proper reduced moment of inertia value, which increases t h e predicted excitation energy by@.
The inclusion of pairing effects would increase this value still further. Perhaps an overall statement about t h e use of the sum rule approach in extracting t h e restoring force constant i s pertinent here.
Unless t h e collective variables are chosen s o a s t o optimal 1 y decouple t h e mode from al1 other degrees of freedom in the system contributions arise in t h e evaluation of t h e restoring force parameter which are unrelated t o t h e mode itself but represent rearrangements of the other variables of t h e system. This can produce serious errors, for example use of t h e contra angular momentum generator rather than (16) leads in t h e oscillator case t o a restoring force constant four times greater than its correct value. The inertial parameter B, it should be said, appears more insensitive t o these changes.
In /9,10/ t h e IBA-2 model has been invoked t o describe t h i s mode.
Such a model i s unable t o make any comment on the excitation energy a s this i s a fit parameter of IBA-2. Its predictions of lower transition rates than other calculations can however be traced directly back t o t h e truncated form of magnetic dipole operator used, one which from the outset neglects al1 but valence contributions. Recent work /Il/ has shown that inclusion of g bosons leads t o fragmentation of t h e state, t h e strength being distributed over a broad energy range. The model is therefore unstable against such augmentations. Regarding form factors, it should perhaps be pointed out that for low momentum transfer, (which sufiice t o describe t h e present experimental- situation), on ignoring normalization factors t h e essential ingredient t o which they are
sensitive i s t h e nuclear radius. Thus most models cited here would give much the same functional dependence of t h e form factor' on t h e energy and there i s therefore nothing particularly special about t h e IBA resul t .
In /12/ a treatment based on t h e vibrating potential model, restricted t o self conjugate nuclei i s given. The generators appropriate t o t h e model, explicitly exhibited in / 8 / , show
surprisingly, that it produces an incoherent collective excited state.
It i s possible, a s has been suggested /13/ that t h e underlying true collective variables are hidden in this formalism. Pairing effects were not taken into account, which in t h i s case would improve the final excitation energy result.
In /14/ a RPA based study i s presented in which both correlations and t h e effects of neutron excess are treated. The calculation was restricted t o a single proton and neutron oscillator shell. An unrealistic aspect however i s that the self consistency conditions require a proton deformation almost 50% greater than that for t h e neutrons.
T o date /15/ represents t h e moet elaborate study of this mode. The treatment i s based on a RPA plus HFB approach in which both
correlations and realistic well shape are included. Three shells for protons and neutrons were used, but al1 A N = 2 coupling terms were neglected, which i s a pity since this precludes the study from making any comparison of t h e collectivity which can be associated with different generators. The distinction between t h e Giant Angle Dipole and pure contra rotational mode i s lost within this truncation scheme.
The final energy qained for ' 5 b ~ d of around 2.4 Mev and a B(M1) strength of 3.2 k s t i l l show substantial dif f erences f rom t h e experimentally observed values. It i s possible that we are seeing signs of t h e fact that the forces used s o succesfully in theories of rotation are insufficient t o describe this new mode.
T o conclude we should like t o emphasise that the Giant Angle Dipole basi cal 1 y requi r e s knowl edge of new i nerti al parameters and restoring forces related t o both rotation and deformation. It thus offers u s a new bridge, connecting dif f erent areas of nuclear structure.
Investigation of t h i s mode should thus provide connections between inertial parameters, inter-nucleon forces and density distributions different from t h e present cnes and thus promises t o open up a rich source of nuclear structure physics.
ACKNOWLEDGEMENTS
Constructive comments from George Bertsch, David Brink, Kenji Hara, Saburo Iwasaki, Tony Lane, Hans Jorg Mang, Peter Ring, and George Ripka are grate+ully acknowledged. Fruitful discussions with D. Bohle and H.M. Sofia are acknowledged with thanks.
REFERENCEÇ
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